Time-to-Event Modeling

• Time-to-event modeling is a statistical framework that quantifies the time until an event occurs while handling censored data and predicting survival and hazard functions.
• Recent advancements integrate classical methods with modern techniques such as deep neural networks, joint modeling of longitudinal data, and federated learning for privacy preservation.
• Model evaluation emphasizes ranking, calibration, and uncertainty quantification, crucial for applications in clinical risk management and predictive maintenance.

Time-to-event modeling, commonly referred to as survival analysis, is the statistical framework focused on quantifying the time duration before a specified event occurs. Key distinguishing features from standard regression include the presence of censored observations (where the true event time is only partially observed) and the goal of predicting distributional quantities, such as survival or hazard functions, rather than single point outcomes. The field has evolved from classical nonparametric and semiparametric procedures to encompass modern, computational approaches that incorporate high-dimensional covariates, multivariate longitudinal data, deep neural networks, and privacy-preserving federated computation.

1. Fundamental Concepts in Time-to-Event Data

Time-to-event data consists of pairs (T, Δ), where T is the event or censoring time and Δ is an event indicator (1 if the event occurred, 0 if censored). Central quantities include the survival function $S(t \mid \mathbf{x}) = \Pr(T > t \mid \mathbf{x})$ and the hazard function $$h(t \mid \mathbf{x}) = \lim_{\Delta t \rightarrow 0} \frac{\Pr(t \leq T < t + \Delta t \mid T \geq t, \mathbf{x})}{\Delta t}$$. The data structure is characterized by censoring, most commonly right-censoring.

Classical nonparametric estimators include the Kaplan–Meier estimator for survival and the Nelson–Aalen estimator for cumulative hazard. The semiparametric Cox proportional hazards model is the canonical regression framework, with the hazard expressed as

$$h(t \mid \mathbf{x}) = h_0(t) \exp(\mathbf{x}^\top \beta)$$

where $h_0(t)$ is an unspecified baseline hazard. Parametric alternatives specify the baseline hazard (e.g., Weibull, log-normal, exponential). Extensions to discrete time—modeling the probability of an event in fixed intervals—are central in many applications and can be formulated as binary response models (Berger et al., 2017).

Time-dependent covariates, non-proportional hazards, competing risks, and the dynamic or recurrent event context add additional structure and complexity.

2. Joint Modeling of Longitudinal and Time-to-Event Data

Contemporary biomedical and reliability studies often measure multiple longitudinal biomarkers over time, with interest focused on their joint association with a subsequent event (Albert et al., 2010, Baghfalaki et al., 7 Dec 2024). Traditional joint models specify a multivariate mixed-effects model for the longitudinal processes and link these to the event model via the “true” (latent, error-free) marker trajectories. A significant technical barrier is high-dimensional integration: with P longitudinal markers, each with random intercept and slope, the joint likelihood requires integration over a $2P$-dimensional random effects vector, quickly becoming computationally intractable.

Two-stage methods have been developed to address scalability. In one approach (Albert et al., 2010), the conditional distribution of longitudinal markers given event time is approximated by fitting all pairwise bivariate mixed models and aggregating estimates. Simulations from these conditional models generate complete pseudo data to account for informative dropout, after which regression calibration is used to mitigate estimation bias in the survival component. A recent extension (Baghfalaki et al., 7 Dec 2024) further improves two-stage methods by fitting one-marker joint models for each marker with its event time and imputing predicted marker trajectories, which are subsequently used as (possibly time-dependent) covariates in a proportional hazards model; a multiple imputation technique propagates uncertainty from marker prediction into the hazard model, yielding robust standard errors and dynamic predictions even with high-dimensional longitudinal panels.

Other advances, such as JLCT (Joint Latent Class Trees) (Zhang et al., 2018), integrate tree-based latent class discovery with flexible survival modeling, allowing time-varying covariates both for class membership and within-class survival/hazard modeling. Shared parameter models have been adapted to incorporate geometric features (e.g., curvature, peak values) of cyclical longitudinal processes measured on nested timescales, substantially broadening the physiological signals that can be connected with time-to-event outcomes (Saha et al., 2021). Hierarchically structured random effects and mixture distributions are critical for representing individual-level heterogeneity, non-ignorable dropout, and latent class structure.

3. Flexible and Modern Statistical Methodologies

The broad toolbox of survival regression has expanded far beyond classical Cox, AFT (accelerated failure time), and additive hazards models:

• RAFT/RPH regression (Sperrin et al., 2011) modifies the standard time scale, using age rather than time since observation, and regresses age-varying covariates on age prior to inclusion in the event model, thereby removing confounding by observation time.
• Semiparametric and tree-based discrete time-to-event regression (Berger et al., 2017) implement smoothing (via penalized splines) for baseline hazard and nonlinear covariate effects, and recursive partitioning (CART) for automatic detection of interactions among covariates and time.
• Vine copula models (Pan et al., 2021) uniquely allow the flexible modeling of the dependence between mixed-type explanatory variables and a (possibly censored) survival response without enforcing proportionality or strict monotonicity in conditional hazards or survival, enabling accurate interval prediction even when standard Cox/AFT assumptions fail.

Treatment effect estimation in the presence of unobserved confounding has motivated bivariate transformation models with structured bivariate Gaussian copulas and penalized maximum likelihood estimation (Marra et al., 21 Oct 2024). Bayesian cure rate models (Papastamoulis et al., 16 Sep 2024) explicitly separate “cured” versus susceptible populations and permit mixture and promotion time modeling for latency distributions, with inference via MCMC and embedded parallel tempering.

Survival function matching (Chapfuwa et al., 2019) and adversarial frameworks (Chapfuwa et al., 2018) utilize neural networks to match the full conditional survival function (or its empirical estimate) nonparametrically. These approaches are distinguished by whether they use explicit adversaries (DATE (Chapfuwa et al., 2018)) or calibration losses based on empirical distribution alignment (SFM (Chapfuwa et al., 2019)), with the key statistical benefit being better-calibrated survival probability and sharper uncertainty quantification compared to models focused on concordance alone.

4. Deep Learning and Neural Time-to-Event Models

Neural models for survival prediction now encompass a spectrum of architectures and objectives (Chen, 1 Oct 2024):

• DeepSurv extends Cox regression by parameterizing the log-risk score with a neural network, trained by the classic partial likelihood (ranking) loss.
• DeepHit parameterizes a discrete-time PMF over time bins, combining negative log-likelihood and ranking loss terms; it generalizes naturally to competing risks and dynamic (time-series) settings.
• Cox-Time introduces time-dependent neural risk scores, thereby relaxing the proportional hazards constraint.
• SODEN and other neural ODE-based approaches parameterize the cumulative hazard as an ODE whose right-hand side is modeled by a neural network, unifying continuous and discrete time by direct hazard integration.
• Kernel survival analysis and deep kernel learning translate nonparametric conditional survival estimation into learned, neural-network-based similarity structures, generalizing nearest neighbors and kernel-based approaches.
• Survival function matching frameworks (Chapfuwa et al., 2019) use a direct (differentiable) calibration loss between model-implied and empirical survival curves, leveraging reparameterization or heuristic gradient estimators instead of adversarial discriminators.

These models, particularly when extended to dynamic inputs via recurrent or transformer modules, permit individualized real-time event probability forecasting, supporting applications in electronic health records, predictive maintenance, and beyond. Multi-task formulations appear increasingly in physiological modeling, where auxiliary tasks such as lab value imputation, identity invariance via adversarial training, and PCGrad-based gradient conflict resolution drive gains in long-horizon early warning systems (Kataria et al., 25 Sep 2025).

5. Federated and Privacy-Preserving Time-to-Event Analysis

Regulatory and privacy constraints in multi-center studies prevent aggregation of individual-level event data. A modern line of research addresses this by introducing federated approaches (Jang et al., 28 Jul 2025), where each site summarizes its survival data through pseudo-observations derived from influence functions or the Kaplan–Meier estimator. Aggregation across sites is achieved via renewable generalized linear modeling of pseudo-values, requiring only local parameter and summary matrix exchange.

Soft-thresholding debiasing methodologies further adapt local coefficient estimates to be shrunk toward global values unless local evidence justifies deviation, thus addressing site-level heterogeneity without inflating model variance. This federated approach supports estimation of survival probabilities with both time-constant and time-varying covariate effects, under both proportional and non-proportional hazards settings, and has demonstrated empirical performance comparable to pooled individual-level analysis.

6. Calibration, Evaluation, and Practical Application

Model performance in time-to-event analysis is evaluated along three axes: ranking (e.g., Concordance Index [CI]), calibration (agreement between predicted and empirical survival probabilities; e.g., Integrated Brier Score [IBS], calibration slope), and concentration/uncertainty (sharpness of predicted distributions). Proper calibration is increasingly emphasized, with neural approaches that explicitly match predicted and observed distributions outperforming methods tuned only for CI (Chapfuwa et al., 2019). Ensemble methods linearly combining several model types can further increase both ranking accuracy and calibration robustness beyond any single model (Fernandez et al., 12 Mar 2024).

Bayesian and evidential frameworks (Huang et al., 19 Jun 2024, Papastamoulis et al., 16 Sep 2024) enrich model outputs with explicit quantification of both aleatory (variance) and epistemic (precision or belief/plausibility) uncertainty. These properties are especially important in clinical or high-stakes settings where risk tolerability and uncertainty aversion direct downstream decision making.

Dynamic landmark and individualized prediction, competing risks, and context-dependent interpretability (by latent class structure, tree partitions, or kernel support) further enhance the practical utility of time-to-event modeling for both retrospective analysis and forward-looking risk management.

7. Current Directions and Future Prospects

Research in time-to-event modeling is focused on:

• Scalable joint modeling of complex (multivariate, multidimensional, and dynamic) longitudinal and event data at scale.
• Calibration-focused training objectives for survival probability estimation.
• Integration of deep learning architectures with established statistical design patterns, e.g., combining neural risk scores with classical partial likelihood or pseudo-observation frameworks.
• Privacy-preserving and federated computation schemes for distributed data environments.
• Flexible modeling frameworks (copulas, mixtures, semiparametric trees) that relax restrictive assumptions and allow for both nonlinear effects and heterogeneous population structures.
• Uncertainty quantification and interpretability, with formal connections to causal inference and statistical guarantees.

Comprehensive code repositories and R packages (e.g., TSJM, jlctree, GJRM) are being released to support replicability and practical adoption, with end-to-end pipelines capable of handling landmark prediction, competing risks, cure fraction inference, and hybrid or federated learning.

The continuing expansion of time-to-event modeling is fueled by methodological innovation, computational advances, and an ever-increasing variety of application domains. This progress positions survival analysis as both a foundational and rapidly evolving component of modern statistical science and machine learning.